Resolving the spurious-state problem in Dirac equation by using the staggered-grid method

TL;DR

This work tackles the spurious-state problem that arises when discretizing the Dirac equation on a uniform grid with central differences. It introduces a staggered-grid finite-difference method (SGM) that places the large and small Dirac components on interlaced nodes and computes derivatives between staggered points, thereby breaking the symmetry between $H_\kappa$ and $H_{-\kappa}$ and eliminating spurious states without Wilson terms. Benchmarking against Woods–Saxon potentials for $^{132}$Sn shows that the 3PSGM and 5PSGM schemes yield energies in close agreement with shooting methods, with 3PNHSGM providing real spectra under a non-Hermitian formulation; the method also preserves correct asymptotic behavior for weakly bound states. The approach offers a simple, Hermitian-compatible framework that can be extended to higher-order and multi-dimensional problems, providing a compact tool for relativistic bound-state and scattering calculations.

Abstract

Discretizing the Dirac equation on a uniform grid with the central difference formula often generates spurious states. We propose a staggered-grid scheme in the framework of the finite-difference method that suppresses these spurious states without introducing Wilson terms or ad-hoc filtering. In this approach, the large and small components of the Dirac equation are placed on interlaced nodes, and the first-order derivatives are evaluated between staggered points, yielding a Hamiltonian that breaks the unitary transformation between $H_κ$ and $H_{-κ}$. Benchmarks with the nuclear Woods-Saxon potentials demonstrate one-to-one agreement with the eigenvalues obtained from shooting method and asymmetric finite-difference method, rapid convergence for weakly bound states, and reduced box-size sensitivity. The method retains the simplicity of central differences and standard matrix diagonalization, while naturally extending to higher-order and multi-dimension systems. It provides a compact and efficient tool for relativistic bound-state and scattering calculations.

Figures (2)

• Figure 1: In the Schematic of SGM, large and small components, $G(r)$ and $F(r)$ are shifted by $h/2$.
• Figure 2: The wave functions $G(r)$ and $F(r)$ of $3p_{1/2}$ in $^{132}$Sn from the 3PSGM, 3PFDM, and shooting method.